Properties

 Base field $$\Q(\sqrt{17})$$ Label 2.2.17.1-576.6-n5 Conductor $$(15 a - 27)$$ Conductor norm $$576$$ CM no base-change no Q-curve yes Torsion order $$4$$ Rank not available

Related objects

Show commands for: Magma / SageMath / Pari/GP

Base field $$\Q(\sqrt{17})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 4$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 4)

gp: K = nfinit(a^2 - a - 4);

Weierstrass equation

$$y^2 + \left(a + 1\right) x y + \left(a + 1\right) y = x^{3} + \left(a + 1\right) x^{2} + \left(271 a - 686\right) x + 3406 a - 8718$$
magma: E := ChangeRing(EllipticCurve([a + 1, a + 1, a + 1, 271*a - 686, 3406*a - 8718]),K);

sage: E = EllipticCurve(K, [a + 1, a + 1, a + 1, 271*a - 686, 3406*a - 8718])

gp: E = ellinit([a + 1, a + 1, a + 1, 271*a - 686, 3406*a - 8718],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(15 a - 27)$$ = $$\left(-a - 1\right)^{6} \cdot \left(3\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$576$$ = $$2^{6} \cdot 9$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(-297 a + 4797)$$ = $$\left(-a - 1\right)^{18} \cdot \left(3\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$21233664$$ = $$2^{18} \cdot 9^{2}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-\frac{14326000}{9} a + \frac{36913625}{9}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E);  sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

magma: Rank(E);

sage: E.rank()

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

Torsion subgroup

Structure: $$\Z/2\Z\times\Z/2\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(4 a - 13 : 2 a - 2 : 1\right)$,$\left(-10 a + 23 : -2 a + 8 : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[3]

Local data at primes of bad reduction

magma: LocalInformation(E);

sage: E.local_data()

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-a - 1\right)$$ $$2$$ $$4$$ $$I_{8}^*$$ Additive $$-1$$ $$6$$ $$18$$ $$0$$
$$\left(3\right)$$ $$9$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ except those listed.

prime Image of Galois Representation
$$2$$ 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 576.6-n consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is not the base-change of an elliptic curve defined over $$\Q$$. It is a $$\Q$$-curve.